An addition type formula for the elliptic digamma function

Abstract

Eisenstein derived addition formulas for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite sum of the cotangent functions. In this paper, we apply this idea of Eisenstein's to the addition type formula for the double cotangent function, established by the author. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl-Kaneko-Zagier.